Distributed lag

In econometrics, a distributed lag model is a time series model in which both the current value of the explanatory variable and the values of this variable in previous periods are included in the regression equation.

The simplest example of a distributed lag model: $y_{t}=a_{0}+b_{0}x_{t}+b_{1}x_{t-1}+\varepsilon _{t}$ ${\ displaystyle y_ {t} = a_ {0} + b_ {0} x_ {t} + b_ {1} x_ {t-1} + \ varepsilon _ {t}}$ ${\ displaystyle y_ {t} = a_ {0} + b_ {0} x_ {t} + b_ {1} x_ {t-1} + \ varepsilon _ {t}}$ . More generally,

$y_{t}=a_{0}+b_{0}x_{t}+b_{1}x_{t-1}+b_{2}x_{t-2}+...+b_{p}x_{t-p}+\varepsilon _{t}$ ${\ displaystyle y_ {t} = a_ {0} + b_ {0} x_ {t} + b_ {1} x_ {t-1} + b_ {2} x_ {t-2} + ... + b_ { p} x_ {tp} + \ varepsilon _ {t}}$ ${\ displaystyle y_ {t} = a_ {0} + b_ {0} x_ {t} + b_ {1} x_ {t-1} + b_ {2} x_ {t-2} + ... + b_ { p} x_ {tp} + \ varepsilon _ {t}}$

Here we can talk about the short-term effect of the explanatory variable on the explainable ( $b_{0}$ ${\ displaystyle b_ {0}}$ $b_ {0}$ ), as well as long-term ( $\sum _{i=0}^{p}b_{i}$ ${\ displaystyle \ sum _ {i = 0} ^ {p} b_ {i}}$ ${\ displaystyle \ sum _ {i = 0} ^ {p} b_ {i}}$ ) This model, in turn, is a special case of the autoregressive model and distributed lag .

Examples of macroeconomic models in which the time lag is important:

Consumption function
Making money in the banking system
The relationship between money supply and price levels
The lag between R&D costs and productivity
J-curve of the relationship between the exchange rate and the trade balance
Investment Accelerator Model

The reasons for the existence of lags can be divided into three groups:

Technological
Institutional
Psychological

The main difficulty for the empirical estimation of a distributed lag model is the presence of multicollinearity , since in economic data the adjacent values of the same data series are usually highly correlated with each other. In addition, it is not always possible to determine a priori how many lag variables it is worth including in the model. There are even models with an infinite number of lag regressions, the coefficients of which are infinitely reduced (for example, exponentially). There are many special technologies for working with distributed lags: for example, the Tinbergen and Alta method is the “thumb method” for determining the optimal number of lag variables without introducing additional prerequisites to the model. The Koyka and Almon models, on the contrary, introduce prerequisites for lag coefficients to simplify their estimation.

Tinbergen and Alta Approach

The Tinbergen and Alta approach allows us to find a balance between the accuracy of the model (the number of included lag variables) and the quality of the estimate (multicollinearity). It involves a consistent assessment of models:

$y_{t}=a_{0}+b_{0}x_{t}+\varepsilon _{t}$ ${\ displaystyle y_ {t} = a_ {0} + b_ {0} x_ {t} + \ varepsilon _ {t}}$
$y_{t}=a_{0}+b_{0}x_{t}+b_{1}x_{t-1}+\varepsilon _{t}$ ${\ displaystyle y_ {t} = a_ {0} + b_ {0} x_ {t} + b_ {1} x_ {t-1} + \ varepsilon _ {t}}$
$y_{t}=a_{0}+b_{0}x_{t}+b_{1}x_{t-1}+b_{2}x_{t-2}+\varepsilon _{t}$ ${\ displaystyle y_ {t} = a_ {0} + b_ {0} x_ {t} + b_ {1} x_ {t-1} + b_ {2} x_ {t-2} + \ varepsilon _ {t} }$
...

Stopping the process is recommended when any of the coefficients of the lag variables changes sign or becomes statistically insignificant, which is a consequence of the occurrence of multicollinearity . In addition, it is unlikely, but a situation is possible where there simply will not be enough observations to further increase the number of lag variables.

Convert Bed

The Bed transformation is a technique that allows you to evaluate a model with distributed lags by simply assuming that the coefficients of the lag variables decrease exponentially with increasing lag:

$y_{t}=a_{0}+\sum _{i=0}^{\infty }b\lambda ^{i}x_{t-i}+\varepsilon _{t}$ ${\ displaystyle y_ {t} = a_ {0} + \ sum _ {i = 0} ^ {\ infty} b \ lambda ^ {i} x_ {ti} + \ varepsilon _ {t}}$

In this model, it is easy to find the average lag ${\frac {\lambda }{1-\lambda }}$ ${\ displaystyle {\ frac {\ lambda} {1- \ lambda}}}$ as well as the median lag $log_{\lambda }{0.5}$ ${\ displaystyle log _ {\ lambda} {0.5}}$ .

Subtracting the equation for $y_{t-1}$ ${\ displaystyle y_ {t-1}}$ times $\lambda$ ${\ displaystyle \ lambda}$ we get a simple model:

$y_{t}=a_{0}+bx_{t}+\lambda y_{t-1}+\varepsilon _{t}-\lambda \varepsilon _{t-1}$ ${\ displaystyle y_ {t} = a_ {0} + bx_ {t} + \ lambda y_ {t-1} + \ varepsilon _ {t} - \ lambda \ varepsilon _ {t-1}}$

This model can easily be estimated by the usual least squares method without loss of degrees of freedom. Here, however, there is autocorrelation of a random term ( $\varepsilon _{t}-\lambda \varepsilon _{t-1}$ ${\ displaystyle \ varepsilon _ {t} - \ lambda \ varepsilon _ {t-1}}$ c $\varepsilon _{t-1}-\lambda \varepsilon _{t-2}$ ${\ displaystyle \ varepsilon _ {t-1} - \ lambda \ varepsilon _ {t-2}}$ ), and, worse, a random term correlates with an explanatory variable $y_{t-1}$ ${\ displaystyle y_ {t-1}}$ . Therefore, to evaluate the model, it is recommended to use the method of instrumental variables or evaluate the original model using the nonlinear least squares method.

The Bunk Transform illustrates the relationship of distributed lag models and autoregressive models. Koyka models correspond to two widely used theoretical approaches to distributed lags: the adaptive expectation model and the partial / stock adjustment model.

Adaptive Expectations Model

The dependent variable is assumed to be a function of the expected value of the explanatory variable. This is typical, for example, for inflation models.

$y_{t}=b_{0}+b_{1}x_{t}^{e}+\varepsilon _{t}$ ${\ displaystyle y_ {t} = b_ {0} + b_ {1} x_ {t} ^ {e} + \ varepsilon _ {t}}$

Expectations are formed as a weighted average of previous expectations and the current value of the variable:

$x_{t}^{e}=(1-\lambda )x_{t-1}^{e}+\lambda x_{t}$ ${\ displaystyle x_ {t} ^ {e} = (1- \ lambda) x_ {t-1} ^ {e} + \ lambda x_ {t}}$

Algebraic manipulations lead to the construction of a model that coincides in shape with the Koyk model:

$y_{t}=\lambda b_{0}+\lambda b_{1}x_{t}+(1-\lambda )y_{t-1}+(\varepsilon _{t}-(1-\lambda )\varepsilon _{t-1})$ ${\ displaystyle y_ {t} = \ lambda b_ {0} + \ lambda b_ {1} x_ {t} + (1- \ lambda) y_ {t-1} + (\ varepsilon _ {t} - (1- \ lambda) \ varepsilon _ {t-1})}$

Partial adjustment model

Partial adjustment model assumes long-term dependence:

$y_{t}^{*}=b_{0}+b_{1}x_{t}+\varepsilon _{t}$ ${\ displaystyle y_ {t} ^ {*} = b_ {0} + b_ {1} x_ {t} + \ varepsilon _ {t}}$

This is typical, for example, of economic growth models where potential output is determined by demand. However, the explained variable cannot instantly adjust to the explanatory changes:

$y_{t}-y_{t-1}=\delta (y_{t}^{*}-y_{t-1})$ ${\ displaystyle y_ {t} -y_ {t-1} = \ delta (y_ {t} ^ {*} - y_ {t-1})}$

Thus, the fundamental difference between partial adjustment models and adaptive expectations is that which variable does not change instantly: explained or explained. However, their functional form is similar: after transformations, we obtain

$y_{t}=\delta b_{0}+\delta b_{1}x_{t}+(1-\delta )y_{t-1}+\delta \varepsilon _{t}$ ${\ displaystyle y_ {t} = \ delta b_ {0} + \ delta b_ {1} x_ {t} + (1- \ delta) y_ {t-1} + \ delta \ varepsilon _ {t}}$

You may notice that here, unlike the model of adaptive expectations, there is no correlation of errors with each other and with the explanatory variable. However, the choice of the model, of course, should be explained not by the convenience of its assessment, but by the theoretical assumptions underlying the phenomenon under study.

Lagi Almon

Pricing Model $y_{t}=a_{0}+\sum _{i=0}^{p}b_{i}x_{t-i}+\varepsilon _{t}$ ${\ displaystyle y_ {t} = a_ {0} + \ sum _ {i = 0} ^ {p} b_ {i} x_ {ti} + \ varepsilon _ {t}}$ , we can assume that the coefficient of the lag variable changes in a certain way smoothly, and approximate it using the polynomial: $b_{i}=c_{0}+\sum _{j=1}^{q}c_{j}i^{j}$ ${\ displaystyle b_ {i} = c_ {0} + \ sum _ {j = 1} ^ {q} c_ {j} i ^ {j}}$ . The linear transformation of variables allows you to evaluate the model using the usual least-squares method, and the number of degrees of freedom, of course, will be greater than when evaluating $b_{i}$ ${\ displaystyle b_ {i}}$ individually, if only q <p.

$y_{t}=a_{0}+\sum _{i=0}^{p}(c_{0}+\sum _{j=1}^{q}c_{j}i^{j})x_{t-i}+\varepsilon _{t}=a_{0}+c_{0}(\sum _{i=1}^{p}x_{t-i})+\sum _{j=1}^{q}c_{j}(\sum _{i=1}^{p}x_{t-i}i^{j})+\varepsilon _{t}=a_{0}+c_{0}z_{0}+\sum _{j=1}^{q}c_{j}z_{j}+\varepsilon _{t}$ {\ displaystyle y_ {t} = a_ {0} + \ sum _ {i = 0} ^ {p} (c_ {0} + \ sum _ {j = 1} ^ {q} c_ {j} i ^ { j}) x_ {ti} + \ varepsilon _ {t} = a_ {0} + c_ {0} (\ sum _ {i = 1} ^ {p} x_ {ti}) + \ sum _ {j = 1 } ^ {q} c_ {j} (\ sum _ {i = 1} ^ {p} x_ {ti} i ^ {j}) + \ varepsilon _ {t} = a_ {0} + c_ {0} z_ {0} + \ sum _ {j = 1} ^ {q} c_ {j} z_ {j} + \ varepsilon _ {t}}

By imposing various restrictions (maximum degree, initial and final conditions) on polynomials, the most satisfactory model can be constructed. However, this approach leaves room for specification errors and subjective fitting of models, since there is no statistical way to determine the required polynomial shape.